Understand how the difference between the mean and median is affected by skew

State how the measures differ in symmetric distributions

State which measure(s) should be used to describe the center of a skewed distribution

How do the various measures of central tendency
compare with each other? For symmetric
distributions, the mean, median, trimean, and trimmed mean
are equal, as is the mode except in bimodal
distributions. Differences among the measures occur with skewed
distributions. Figure 1 shows the distribution of 642 scores on
an introductory psychology test. Notice this distribution has
a slight positive skew.

Figure 1. A distribution with a positive
skew.

Measures of central tendency are shown in Table
1. Notice they do not differ greatly, with the exception that
the mode is considerably lower than the other measures. When
distributions have a positive skew,
the mean is typically higher than the median, although it may
not be in bimodal distributions. For these data, the mean of
91.58 is higher than the median of 90. Typically the trimean
and trimmed
mean will fall between the median
and the mean,
although in this case, the trimmed mean is slightly lower than
the median. The geometric
mean is lower than all measures except the mode.

Table 1. Measures of central tendency

for the test scores.

Measure

Value

Mode
Median
Geometric Mean
Trimean
Mean trimmed 50%
Mean

84.00
90.00
89.70
90.25
89.81
91.58

The distribution of baseball salaries (in 1994)
shown in Figure 2 has a much more pronounced skew than the distribution
in Figure 1.

Figure 2. A distribution with a very large positive skew. This histogram shows the salaries of major league baseball players (in thousands of dollars: 25 equals 250,000).

Table 2 shows the measures of central
tendency for these data. The large skew results in very
different values for these measures. No single measure of
central tendency is sufficient for data such as these. If
you were asked the very general question: "So, what
do baseball players make?"
and answered with the mean of $1,183,000, you would not have
told the whole story since only about one third of baseball
players make that much. If you answered with the mode of $250,000
or the median of $500,000, you would not be giving any indication
that some players make many millions of dollars. Fortunately,
there is no need to summarize a distribution with a single number.
When the various measures differ, our opinion is that you should
report the mean, median, and either the trimean or the mean
trimmed 50%. Sometimes it is worth reporting the mode as well.
In the media, the median is usually reported to summarize the
center of skewed distributions. You will hear about median salaries
and median prices of houses sold, etc. This is better than reporting
only the mean, but it would be informative to hear more statistics.